Simplify square root of 2x^3y fourth root of 8xy^7

Solution:

Step:1

Re-express $2x^{3}y$ as $x^{2} \cdot (2xy)$.

Step:1.1

Extract $x^{2}$ from the expression. $\sqrt{x^{2} \cdot (2xy)} \sqrt[4]{8xy^{7}}$

Step:1.2

Switch the positions of $2$ and $x^{2}$. $\sqrt{2 \cdot x^{2}xy} \sqrt[4]{8xy^{7}}$

Step:1.3

Enclose $2xy$ with parentheses. $\sqrt{x^{2} \cdot (2xy)} \sqrt[4]{8xy^{7}}$

Step:1.4

Enclose $2xy$ within parentheses again for clarity. $\sqrt{x^{2} \cdot (2xy)} \sqrt[4]{8xy^{7}}$

Step:2

Extract the $x$ term from under the square root. $x\sqrt{2xy} \sqrt[4]{8xy^{7}}$

Step:3

Rewrite $8xy^{7}$ as $y^{4} \cdot (8xy^{3})$.

Step:3.1

Extract $y^{4}$ from the expression. $x\sqrt{2xy} \sqrt[4]{y^{4} \cdot (8xy^{3})}$

Step:3.2

Shift $x$ to a different position. $x\sqrt{2xy} \sqrt[4]{8y^{4}xy^{3}}$

Step:3.3

Switch the positions of $8$ and $y^{4}$. $x\sqrt{2xy} \sqrt[4]{y^{4} \cdot (8xy^{3})}$

Step:3.4

Enclose $8xy^{3}$ with parentheses. $x\sqrt{2xy} \sqrt[4]{y^{4} \cdot (8xy^{3})}$

Step:3.5

Enclose $8xy^{3}$ within parentheses again for emphasis. $x\sqrt{2xy} \sqrt[4]{y^{4} \cdot (8xy^{3})}$

Step:4

Extract the $y$ term from under the fourth root. $x\sqrt{2xy} (y\sqrt[4]{8xy^{3}})$

Step:5

Multiply $x\sqrt{2xy} (y\sqrt[4]{8xy^{3}})$.

Step:5.1

Convert the expression to have a common index of $4$.

Step:5.1.1

Use the radical conversion $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2xy}$ as $((2xy)^{\frac{1}{2}})$. $x(y((2xy)^{\frac{1}{2}} \sqrt[4]{8xy^{3}}))$

Step:5.1.2

Convert $((2xy)^{\frac{1}{2}})$ to $((2xy)^{\frac{2}{4}})$. $x(y((2xy)^{\frac{2}{4}} \sqrt[4]{8xy^{3}}))$

Step:5.1.3

Rewrite $((2xy)^{\frac{2}{4}})$ as $\sqrt[4]{((2xy)^{2})}$. $x(y(\sqrt[4]{(2xy)^{2}} \sqrt[4]{8xy^{3}}))$

Step:5.2

Combine radicals using the product rule. $x(y\sqrt[4]{(2xy)^{2} (8xy^{3})})$

Step:5.3

Apply the product rule to $2xy$. $x(y\sqrt[4]{(2x)^{2}y^{2} \cdot 8xy^{3}})$

Step:5.4

Combine $y^{2}$ and $y^{3}$ by adding their exponents.

Step:5.4.1

Shift $y^{3}$. $x(y\sqrt[4]{(2x)^{2} (y^{3}y^{2}) \cdot 8x})$

Step:5.4.2

Combine exponents using the power rule $a^{m}a^{n} = a^{m+n}$. $x(y\sqrt[4]{(2x)^{2}y^{3+2} \cdot 8x})$

Step:5.4.3

Sum the exponents $3$ and $2$. $x(y\sqrt[4]{(2x)^{2}y^{5} \cdot 8x})$

Step:5.5

Apply the product rule to $2x$. $x(y\sqrt[4]{2^{2}x^{2}y^{5} \cdot 8x})$

Step:5.6

Square $2$. $x(y\sqrt[4]{4x^{2}y^{5} \cdot 8x})$

Step:5.7

Combine the exponents.

Step:5.7.1

Multiply $8$ by $4$. $x(y\sqrt[4]{32x^{2}y^{5}x})$

Step:5.7.2

Raise $x$ to the power of $1$. $x(y\sqrt[4]{32(x^{1}x^{2})y^{5}})$

Step:5.7.3

Combine exponents using the power rule $a^{m}a^{n} = a^{m+n}$. $x(y\sqrt[4]{32x^{1+2}y^{5}})$

Step:5.7.4

Sum the exponents $1$ and $2$. $x(y\sqrt[4]{32x^{3}y^{5}})$

Step:6

Rewrite $32x^{3}y^{5}$ as $(2y)^{4} \cdot (2x^{3}y)$.

Step:6.1

Factor $16$ out of $32$. $x(y\sqrt[4]{16(2)x^{3}y^{5}})$

Step:6.2

Express $16$ as $2^{4}$. $x(y\sqrt[4]{2^{4} \cdot 2x^{3}y^{5}})$

Step:6.3

Extract $y^{4}$. $x(y\sqrt[4]{2^{4} \cdot 2x^{3}(y^{4}y)})$

Step:6.4

Shift $x^{3}$. $x(y\sqrt[4]{2^{4} \cdot 2y^{4}x^{3}y})$

Step:6.5

Shift $2$. $x(y\sqrt[4]{2^{4}y^{4} \cdot 2x^{3}y})$

Step:6.6

Express $2^{4}y^{4}$ as $(2y)^{4}$. $x(y\sqrt[4]{(2y)^{4} \cdot 2x^{3}y})$

Step:6.7

Enclose with parentheses. $x(y\sqrt[4]{((2y)^{4}) \cdot (2x^{3}y)})$

Step:6.8

Enclose with parentheses for clarity. $x(y\sqrt[4]{((2y)^{4}) \cdot (2x^{3}y)})$

Step:7

Extract terms from under the fourth root. $x(y(2y\sqrt[4]{2x^{3}y}))$

Step:8

Final simplification of the expression.

Step:8.1

Apply the commutative property of multiplication. $2x(y \cdot y\sqrt[4]{2x^{3}y})$

Step:8.2

Multiply $y$ by $y$. $2xy^{2}\sqrt[4]{2x^{3}y}$

Knowledge Notes:

To solve the given problem, we used several mathematical principles and rules:

1. Radical Simplification: Radicals can be simplified by extracting factors that are perfect powers of the radical index.

2. Commutative Property of Multiplication: This property states that the order in which two numbers are multiplied does not affect the product, i.e., $ab = ba$.

3. Power Rule for Exponents: When multiplying like bases, add the exponents, i.e., $a^{m}a^{n} = a^{m+n}$.

4. Radical Conversion: A radical can be converted to an exponent form, i.e., $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$.

5. Product Rule for Radicals: The product of two radicals with the same index can be combined under one radical, i.e., $\sqrt[n]{a}\sqrt[n]{b} = \sqrt[n]{ab}$.

6. Factoring: This involves expressing a number or expression as a product of its factors.

7. Least Common Index: When dealing with multiple radicals, it can be helpful to rewrite them with a common index to combine them more easily.

8. Parentheses: Parentheses are used to clarify the order of operations and group terms together.

By applying these rules systematically, we were able to simplify the given expression step by step.